Chapter S:IV. IV. Informed Search

Transcription

1 Chapter S:IV IV. Informed Search Best-First Search Best-First Search for State-Space Graphs Cost Functions for State-Space Graphs Evaluation of State-Space Graphs Algorithm A* BF* Variants Hybrid Strategies S:IV-110 Informed Search STEIN/LETTMANN

2 BF* Variants For trees G: Breadth-first search is a special case of A*, where h = 0 and c(n, n ) = 1 for all successors n of n. S:IV-111 Informed Search STEIN/LETTMANN

3 BF* Variants For trees G: Breadth-first search is a special case of A*, where h = 0 and c(n, n ) = 1 for all successors n of n. s Node on OPEN 1 1 Node on CLOSED Solved rest problem S:IV-112 Informed Search STEIN/LETTMANN

4 BF* Variants For trees G: Breadth-first search is a special case of A*, where h = 0 and c(n, n ) = 1 for all successors n of n. 1 f = 0 s 1 Node on OPEN Node on CLOSED Solved rest problem f = 1 f = f = 2 f = 2 f = 2 f = 2 f = 3 S:IV-113 Informed Search STEIN/LETTMANN

5 BF* Variants For trees G: Breadth-first search is a special case of A*, where h = 0 and c(n, n ) = 1 for all successors n of n. f = 0 s Node on OPEN 1 1 Node on CLOSED f = 1 f = 1 Solved rest problem f = 2 f = 2 f = 2 f = 2 f = 3 Proof (sketch) 1. g(n) defines the depth of n (consider path from n to s). 2. f(n) = g(n). 3. Breadth-first search the depth difference of nodes on OPEN is Assumption: Let n 1, n 2 be on OPEN, having a larger depth difference: f(n 2 ) f(n 1 ) > For the direct predecessor n 0 of n 2 holds: f(n 0 ) = f(n 2 ) 1 > f(n 1 ). 6. n 1 must have been expanded before n 0 (consider minimization of f under A*). 7. n 1 must have been deleted from OPEN. Contradiction to 4. S:IV-114 Informed Search STEIN/LETTMANN

6 BF* Variants For trees G: Uniform-cost search is a special case of A*, where h = 0. Proof (sketch) See lab class. S:IV-115 Informed Search STEIN/LETTMANN

7 BF* Variants For trees G: Depth-first search is a special case of Z*, where f(n ) = f(n) 1, f(s) = 0, for all successors n of n. S:IV-116 Informed Search STEIN/LETTMANN

8 BF* Variants For trees G: Depth-first search is a special case of Z*, where f(n ) = f(n) 1, f(s) = 0, for all successors n of n. s Node on OPEN -1-1 Node on CLOSED Solved rest problem S:IV-117 Informed Search STEIN/LETTMANN

9 BF* Variants For trees G: Depth-first search is a special case of Z*, where f(n ) = f(n) 1, f(s) = 0, for all successors n of n. -1 f = 0 s -1 Node on OPEN Node on CLOSED Solved rest problem f = -1 f = f = -2 f = -2 f = -3 S:IV-118 Informed Search STEIN/LETTMANN

10 BF* Variants For trees G: Depth-first search is a special case of Z*, where f(n ) = f(n) 1, f(s) = 0, for all successors n of n. f = 0 s Node on OPEN -1-1 Node on CLOSED f = -1 f = -1 Solved rest problem f = -2 f = -2 f = -3 Proof (sketch) 1. f(n ) < f(n) n was inserted on OPEN after n. f(n ) f(n) n was inserted on OPEN after n. 2. Depth-first search the most recently inserted node on OPEN is expanded. 3. Let n 2 be the most recently inserted node on OPEN. 4. Assumption: Let n 1 have been expanded before n 2 f(n 1 ) f(n 2 ). 5. f(n 1 ) < f(n 2 ) (consider minimization of f under Z*). 6. n 1 was inserted on OPEN after n n 2 is not the most recently inserted node on OPEN. Contradiction to 3. S:IV-119 Informed Search STEIN/LETTMANN

11 BF* Variants OPEN List Restriction: Hill-Climbing (HC) Hill-climbing is an informed, irrevocable search strategy. HC characteristics: local or greedy optimization: take the direction of steepest ascend (sometimes: descend) never look back : alternatives are not remembered no OPEN/CLOSED lists usually low computational effort a strategy that is often applied by humans s s γ γ S:IV-120 Informed Search STEIN/LETTMANN

12 BF* Variants Algorithm: Input: Output: HC s. Start node representing the initial problem. successors(n). Returns the successors of node n. (n). Predicate that is True if n is a goal node. f(n). Evaluation function for a node n. A goal node or the symbol Fail. S:IV-121 Informed Search STEIN/LETTMANN

13 Hill-Climbing [ DFS] [BT] Algorithm: Input: Output: HC s. Start node representing the initial problem. successors(n). Returns the successors of node n. (n). Predicate that is True if n is a goal node. f(n). Evaluation function for a node n. A goal node or the symbol Fail. HC(s, successors,, f) 1. n = s; 2. n opt = s; 3. LOOP 4. IF (n) THEN RETURN(n); 5. FOREACH n IN successors(n) DO // Expand n. add_backpointer(n, n); IF (f(n ) > f(n opt )) THEN n opt = n ; // Remember optimum successor. ENDDO 6. IF (n opt = n) THEN RETURN(Fail); // We could not improve. ELSE n = n opt ; // Continue with the best successor. 7. ENDLOOP S:IV-122 Informed Search STEIN/LETTMANN

14 BF* Variants HC Discussion HC issue: The first property of a systematic control strategy, Consider all objects in S., is violated by hill-climbing if no provisions are made. The forecast of the evaluation function (cost function, merit function) may be at least sometimes wrong and misguiding the search. Search will probably terminate at a local optimum. Alternative paths are not considered since each step is irrevocable. S:IV-123 Informed Search STEIN/LETTMANN

15 BF* Variants HC Discussion HC issue: The first property of a systematic control strategy, Consider all objects in S., is violated by hill-climbing if no provisions are made. The forecast of the evaluation function (cost function, merit function) may be at least sometimes wrong and misguiding the search. Search will probably terminate at a local optimum. Alternative paths are not considered since each step is irrevocable. Workaround: Perform multiple restarts (e.g. random-restart hill climbing). Workaround issue: The second property of a systematic control strategy, Consider each object in S only once., is violated if no provisions are made. S:IV-124 Informed Search STEIN/LETTMANN

16 BF* Variants HC Discussion (continued) Hill-climbing can be the favorite strategy in certain situations: (a) We are given a highly informative evaluation function to control search. (b) The operators are commutative. Commutativity is given, if all operators are independent of each other. The application of an operator will 1. neither prohibit the applicability of any other operator, 2. nor modify the outcome of its application. Example: Expansion of the nodes in a complete graph. S:IV-125 Informed Search STEIN/LETTMANN

17 Remarks: Given commutativity, an irrevocable search strategy can be applied without hesitation: finding the optimum may be postponed but is never prohibited. Keywords: greedy algorithm, greedy strategy, matroid Given commutativity, hill-climbing can be considered a systematic strategy. Typically, hill-climbing is operationalized as an informed strategy, i.e., information about the goal (or about a concept to reach the goal) is exploited. If such external or look-ahead information is not exploited, hill-climbing must be considered an uninformed strategy. Q. What could be a provision to avoid a violation of the second property of a systematic control strategy? S:IV-126 Informed Search STEIN/LETTMANN

18 BF* Variants OPEN List Restriction: Best-First Beam Search [Rich & Knight 1991] Characteristics: Best-first search is used with an OPEN list of limited size k. If OPEN exceeds its size limit, nodes with worst f-values are discarded until size limit is adhered to. Operationalization: 1. A cleanup_closed function is needed to prevent CLOSED from growing uncontrollably. S:IV-127 Informed Search STEIN/LETTMANN

19 Remarks: For k = 1 this is identical to an hill-climbing search. In breadth-first beam search [Lowerre 1976] all (at most) k nodes of the current level are expanded and only the best k of all these successors are kept and used for the next level. S:IV-128 Informed Search STEIN/LETTMANN

20 Hybrid Strategies Spectrum of Search Strategies The search strategies Hill-climbing irrevocable decisions, consideration of newest alternatives Informed backtracking tentative decisions, consideration of newest alternatives Best-first search tentative decisions, consideration of all alternatives form the extremal points within the spectrum of search strategies, based on the following dimensions: R Recovery. How many previously suspended alternatives (nodes) are reconsidered after finding a dead end? S Scope. How many alternatives (nodes) are considered for each expansion? S:IV-129 Informed Search STEIN/LETTMANN

21 Hybrid Strategies Spectrum of Search Strategies The search strategies Hill-climbing irrevocable decisions, consideration of newest alternatives Informed backtracking tentative decisions, consideration of newest alternatives Best-first search tentative decisions, consideration of all alternatives form the extremal points within the spectrum of search strategies, based on the following dimensions: R Recovery. How many previously suspended alternatives (nodes) are reconsidered after finding a dead end? S Scope. How many alternatives (nodes) are considered for each expansion? S:IV-130 Informed Search STEIN/LETTMANN

22 Hybrid Strategies Spectrum of Search Strategies Scope: Amount of alternatives considered for each expansion S Consideration of all alternatives Consideration of only newest alternatives Irrevocable decisions Tentative decisions R Recovery: Amount of suspended alternatives reconsidered in dead end situations The large scope of best-first search requires a high memory load. This load can be reduced by mixing it with backtracking. S:IV-131 Informed Search STEIN/LETTMANN

23 Hybrid Strategies Spectrum of Search Strategies Scope: Amount of alternatives considered for each expansion S Best-First Search Consideration of all alternatives Hill-Climbing Consideration of only newest alternatives Irrevocable decisions Tentative decisions Backtracking R Recovery: Amount of suspended alternatives reconsidered in dead end situations The large scope of best-first search requires a high memory load. This load can be reduced by mixing it with backtracking. S:IV-132 Informed Search STEIN/LETTMANN

24 Hybrid Strategies Spectrum of Search Strategies Scope: Amount of alternatives considered for each expansion S Best-First Search Consideration of all alternatives Hill-Climbing Consideration of only newest alternatives Irrevocable decisions Tentative decisions Backtracking R Recovery: Amount of suspended alternatives reconsidered in dead end situations The large scope of best-first search requires a high memory load. This load can be reduced by mixing it with backtracking. S:IV-133 Informed Search STEIN/LETTMANN

25 Remarks: Recall that the memory consumption of best-first search is an (asymptotically) exponential function of the search depth. Hill-climbing is the most efficient strategy, but its effectiveness (solution quality) can only be guaranteed for problems that can be solved with a greedy approach. Informed backtracking requires not as much memory as best-first search, but usually needs more time as its scope is limited. Without a highly informed heuristic h, the degeneration of best-first strategies down to a uniform-cost search is typical and should be expected as the normal case. S:IV-134 Informed Search STEIN/LETTMANN

26 Hybrid Strategies Strategy 1: BF at Top s Characteristics: Best-first search is applied at the top of the search space graph. Backtracking is applied at the bottom of the search space graph. Operationalization: 1. Best-first search is applied until a memory allotment of size M 0 is exhausted. 2. Then backtracking starts with a most promising node n on OPEN. 3. If backtracking fails, it restarts with the next most promising OPEN node. S:IV-135 Informed Search STEIN/LETTMANN

27 Hybrid Strategies Strategy 2: BF at Bottom s d 0 Characteristics: Backtracking is applied at the top of the search space graph. Best-first search is applied at the bottom of the search space graph. Operationalization: 1. Backtracking is applied until the search depth bound d 0 is reached. 2. Then best-first search starts with the node at depth d If best-first search fails, it restarts with the next node at depth d 0 found by backtracking. S:IV-136 Informed Search STEIN/LETTMANN

28 Remarks: The depth bound d 0 in Strategy 2 must be chosen carefully in order to avoid that the best-first search does not run out of memory. Hence, this strategy is more involved than Strategy 1 where the switch between best-first search and backtracking is triggered by the exhausted memory. If a sound depth bound d 0 is available, Strategy 2 (best-first search at bottom) is usually superior to Strategy 1 (best-first search at top). Q. Why? S:IV-137 Informed Search STEIN/LETTMANN

29 Hybrid Strategies Strategy 3: Extended Expansion s Characteristics: Best-first search acts locally to generate a restricted number of promising nodes. Informed depth-first search acts globally, using best-first as an extended node expansion. Operationalization: 1. An informed depth-first search selects the nodes n for expansion. 2. But a best-first search with a memory allotment of size M 0 is used to expand n. 3. The nodes on OPEN are returned to the depth-first search as direct successors of n. S:IV-138 Informed Search STEIN/LETTMANN

30 Hybrid Strategies Strategy 3: Extended Expansion s Characteristics: Best-first search acts locally to generate a restricted number of promising nodes. Informed depth-first search acts globally, using best-first as an extended node expansion. Operationalization: 1. An informed depth-first search selects the nodes n for expansion. 2. But a best-first search with a memory allotment of size M 0 is used to expand n. 3. The nodes on OPEN are returned to the depth-first search as direct successors of n. S:IV-139 Informed Search STEIN/LETTMANN

31 Remarks: Strategy 3 is an informed depth-first search whose node expansion is operationalized via a memory-restricted best-first search. Q. What is the asymptotic memory consumption of Strategy 3 in relation to the search depth? S:IV-140 Informed Search STEIN/LETTMANN

32 Hybrid Strategies Strategy 4: IDA* [Korf 1985] Characteristics: Depth-first search is used in combination with an iterative deepening approach for f-values. Nodes are considered only if their f-values do not exceed a given threshold. Operationalization: 1. limit is initialized with f(s). 2. In depth-first search, only nodes are considered with f(n) limit. 3. If depth-first search fails, limit is increased to the minimum cost of all f-values that exceeded the current threshold and depth-first search is rerun. S:IV-141 Informed Search STEIN/LETTMANN

33 Remarks: IDA* always finds a cheapest solution path if the heuristic is admissible, or in other words never overestimates the actual cost to a goal node. IDA* uses space linear in the length of a cheapest solution. IDA* expands the same number of nodes, asymptotically, as A* in an exponential tree search. S:IV-142 Informed Search STEIN/LETTMANN

34 Hybrid Strategies Strategy 5: Focal Search [Ibaraki 1978] Characteristics: An informed depth-first search is used as basic strategy. Nodes are selected from newly generated nodes and the best nodes encountered so far. Operationalization: The informed depth-first search expands the cheapest node n from its list of alternatives. For the next expansion, it chooses from the newly generated nodes and the k best nodes (without n) from the previous alternatives. S:IV-143 Informed Search STEIN/LETTMANN

35 Remarks: For k = 0 this is identical to an informed depth-first search. For k = this is identical to a best-first search. Memory consumption (without proof): O(b d k+1 ), where b denotes the branching degree and d the search depth. An advantage of Strategy 5 is that its memory consumption can be controlled via the single parameter k. Differences to beam search: In focal search no nodes are discarded. Therefore, focal search will never miss a solution. In best-first beam search the OPEN list is of limited size. S:IV-144 Informed Search STEIN/LETTMANN

36 Hybrid Strategies Strategy 6: Staged Search [Nilson 1971] s Characteristics: Best-first search acts locally to generate a restricted number of promising nodes. Hill-climbing acts globally, but by retaining a set of nodes. Operationalization: 1. Best-first search is applied until a memory allotment of size M 0 is exhausted. 2. Then only the cheapest OPEN nodes (and their pointer-paths) are retained. 3. Best-first search continues until Step 1. is reached again. S:IV-145 Informed Search STEIN/LETTMANN

37 Remarks: Staged search can be considered as a combination of best-first search and hill-climbing. While a pure hill-climbing discards all nodes except one, staged search discards all nodes except a small subset. Staged search addresses the needs of extreme memory restrictions and tight runtime bounds. Recall that the Strategies 1-5 are complete with regard to recovery, but that Strategy 6, Hill Climbing, and Best-First Beam Search are not. S:IV-146 Informed Search STEIN/LETTMANN